Question

The receiver in an optical communications system uses a photodetector that counts the number of photons that arrive during one time unit. Suppose that the number X of photons can be modeled as a Poisson random variable with rate λ1 when a signal is present (say bit “1” is transmitted) and a Poisson random variable with rate λ0 < λ1 when a signal is absent (say bit “0” is transmitted). Let p denote the probability that the transmitted bit is “1”. 1. What is the probability that a bit “1” was transmitted if the receiver detects k photons? 2. To decide the value of the transmitted bit, the receiver implements the following decision rule, called Maximum Likelihood (ML) detection: if P[bit = 1|X = k] > P[bit = 0|X = k], decide bit “1” was sent, otherwise bit “0”. Find the corresponding threshold decision rule based on the value of X. 3. What is the probability of error for the above decision rule.

Answer 1

Let the counter follow:

where denotes the bit sent.

Now, we need to find:

For the ML detection rule, we need to have:

And this is the threshold region.

Now let . Then the error is: